Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Inverting Consequences of Uniform Convergence

  1. Jan 17, 2010 #1
    Hi. Now you probably know that if a function fk(x) converges uniformly to f(x) then we are allowed to certain actions such as

    [tex]limn-> \inf \int f (of k) dx = \int f dx [/tex]

    In other words we are allowed to exchange limit and integral. Now say we have any sequnce valued function fk(x) . And we also have

    [tex]limn-> \inf \int f (of k) dx = 0 [/tex]

    Does that imply that fk(x) converges uniformly to 0?
     
  2. jcsd
  3. Jan 17, 2010 #2

    Landau

    User Avatar
    Science Advisor

    Please learn Latex properly.

    Indeed, if [tex](f_k)_{k\in\mathbb{N}}[/tex] converges uniformly to [tex]f[/tex] on some interval [tex][a,b]\subset\mathbb{R}[/tex], then [tex]\lim_{k\to\infty}\int_a^b f_k(x)dx=\int_a^b \lim_{k\to\infty}f_k(x)dx=\int_a^b f(x)dx[/tex].

    The converse of this is:

    if [tex]\lim_{k\to\infty}\int_a^b f_k(x)dx=\int_a^b f(x)dx[/tex], does it follow that [tex](f_k)_{k\in\mathbb{N}}[/tex] converges uniformly to [tex]f[/tex]?

    This is not what you asked, after all [tex]\int_a^b f(x)dx=0[/tex] does not imply [tex]f=0[/tex] (i.e. f(x)=0 for all x).

    The converse is therefore obviously not true. Take for example [a,b]=[0,1], and define f_k and f on [0,1] by f_k(x)=1/k and f(x)=-1 if x<1/2, f(x)=+1 if x>1/2. Then

    [tex]\lim_{k\to\infty}\int_0^1 f_k(x)dx=\lim_{k\to\infty}\frac{1}{k}=0=-\frac{1}{2}+\frac{1}{2}=\int_0^1 f(x)dx[/tex]

    but (f_k)_k does not even converge to f pointwise.

    For an counter-example to your claim, just take (f_k)_k to be a constant sequence: f_k=f as defined above, for all k. Then the integral of f_k is equal to zero (and hence the limit of the integral too), but f_k does not converge (even pointwise) to the zero function.
     
    Last edited: Jan 17, 2010
  4. Jan 17, 2010 #3
    Thanks a lot. And will do.
     
  5. Jan 17, 2010 #4

    Landau

    User Avatar
    Science Advisor

    You're welcome. You can just click on any equation to see its Latex code.
     
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook